MORE TENNIS PHYSICS

New: Hawk-Eye footprints (Added Feb 2014)

1. PowerPoint Presentations

The following files can be viewed directly as PowerPoint presentations or downloaded as .ppt files. They are designed mainly to assist students wanting ideas for tennis projects.

Ball bounce (or download BouncingBalls.ppt about 100 kb)

The sweet spot (or download SweetSpot.ppt about 300 kb)

More advanced projects for University students can be found in the list of publications and in the additional information provided below.

2. Measuring Ball Properties

This is a photo of the Stevens machine owned by Tennis Australia. It is still used in some countries to measure ball stiffness as specified by the rules of tennis. It includes a circular hole in each side to test whether the ball is too large, too small or just right (according to the rules). The test procedure involves several steps:

(1) Squash the ball a few times in a few directions to make sure it is round.

(2) Apply a force equivalent to a 8.165 kg or 18 lb mass sitting on the ball.

(3) Wait 5 seconds then measure the deformation under this load (called “forward deformation”)

(4) Continue to compress the ball until its initial diameter has been reduced by one inch.

(5) Remove this compression until the load on the ball is again 8.165 kg

(6) Wait 10 seconds then measure the deformation under this load (called “return deformation”)

The ITF web site has a movie showing the modern way to do it (using a robot). The simplest way to do it at a University is to ask one of your Engineering Departments to help you set up one of their small materials testing machines to record the force and compression on whatever ball you want to test. They will probably be happy to help you since they normally test less interesting objects like blocks of concrete or metal rods.

BALL TEST RESULTS (July 2007)

Ball test results conducted in my lab, plus wear tests on court, can be found at

http://www.racquetsportsindustry.com/articles/2007/07/ball_testing.html

If you click on the authors names at the bottom of this article you will discover lots of other interesting articles on the technical aspects of tennis, all published in RSI magazine, but you won’t find any photos or stories about famous players or their rankings. You will only find useful information in RSI magazine.

3. Measuring String Properties

There are hundreds of different tennis strings on the market that players can choose from. Exactly why some players prefer one string to another is a bit of a mystery, but a physical measurement of the properties of different strings provides some of the answers. There are only four basic types of string, namely natural gut, nylon, polyester and kevlar in order of increasing stiffness. Stiffness here refers to how far a string will stretch longitudinally (lengthwise) when it is pulled to a given tension. Tennis strings are a lot stiffer than rubber tubes and a lot softer than steel guitar strings.

When Luxilon strings appeared on the market, players noticed an increase in ball spin. It took a while for manufacturers to realise that slippery strings generate more spin than rough textured strings. These days, most top players prefer to use polyester strings since they a more slippery and snap back into place at the end of each shot, giving the ball a bit more spin. That is essentially how spaghetti strings work.

When a string is installed in a tennis racquet, it is pulled to a desired tension, typically about 60 lb or about 28 kg and then tied off at both ends. As soon as the string is tied off, the tension starts dropping dramatically at first and then more slowly as the days and weeks pass by. The tension also drops slightly every time the racquet is used to hit a ball. Some of the equipment I have used to pull and hit strings and to measure how far they stretch lengthwise or sideways, is shown below. I use a sample of string about 40 cm long clamped at each end between metal jaws, with a load cell at one end to measure the tension when the string is stretched either lengthwise or sideways. A laser beam passing through a 1 mm grid is used to measure the sideways stretch of the string when it is impacted with a hammer mounted as a pendulum. The 1mm grid can be seen in the photo attached to the back end of the hammer. The sideways force on the string is roughly equivalent to a 100 mph serve.

Hammer pulled and held in position ready to slam sideways into the string. The string is stretched lengthwise by means of a screw thread attached to the blue clamp at the left hand end of the string.

Red clamp at right hand end of string, attached to a load cell. As soon as the blue clamp is locked into a fixed position, the tension starts dropping rapidly. When the hammer hits the string, the tension increases rapidly and then drops back to its pre-impact value as soon as the hammer bounces off the string. The hammer bounces off the string at about 95% of its incident speed, regardless of string tension, string type or string diameter. All tennis strings are essentially equal in this respect.

Measure String Tension ON-LINE!

Even though a racquet stringer can string your racquet at any tension you like, the actual string tension will be different. String tension drops rapidly as soon as the string is tied off. Furthermore, a huge force is applied to the frame as each new string is added, pulling the frame out of shape and affecting the tension of the strings already installed. The tension drops over time until it is time for a restring.

Players like to hit their strings to hear the “ping” sound to determine whether the strings are tight or loose. The ping frequency of the whole stringbed is essentially the same as the vibration frequency of a single string with a length equal to the square root of the area of the stringbed. For example, if area = 650 sq cm then length = 25.5 cm. QuickTime sound files for a 32 cm long string at various tensions are attached. You can use these files to compare with the ping frequency of your own racquet. Best results (the best “pings”) are obtained by bouncing a golf ball off the strings, with string dampeners removed. The tension in your racquet won’t be the same as the tension in the 32 cm string since the vibration frequency also depends on string diameter and head area, but the result will be roughly correct. You can easily proportion the results since the ping frequency is inversely proportional to string length. A 16 cm long string will vibrate at twice the frequency of a 32 cm long string of the same diameter at the same tension.

Alternatively, you can use the sound files to determine if the string tension has dropped since the last time you tested them. In any given racquet, the ping frequency of the stringbed is proportional to the square root of the string tension, regardless of the string diameter and head size. But you can’t use this technique to compare the string tension in your racquet with another racquet unless you take into account the different head size, string diameter and string type. For example, polyester is denser than nylon so a polyester string will vibrate at a lower frequency than a nylon string of the same diameter and length and tension.

MOVIE FILES

The movies below are QuickTime movies. If you don’t have QuickTime it is available for free for both Mac and Window systems at http://www.apple.com/quicktime/download/mac.html

The 10Mb Balance and Swing Weight movies might take a minute or two to download.

4. Measuring Balance & Swing Weight

The balance point of a racquet is the distance from the butt end of the handle to its centre of mass or its centre of gravity (an old-fashioned term).

Swing Weight is a term commonly used in sports to denote the moment of inertia of a bat or club or racquet. It can be measured by mounting the implement as a pendulum and timing the period for one oscillation. A more accurate result is obtained by timing 10 oscillations and then divide the result by 10. The parallel axis theorem can be used to work out the swing weight about any axis parallel to the one shown in the movie (as described in the Physics and Technology of Tennis book). In tennis, swing weight refers to the moment of inertia about an axis 10 cm from the butt end, and is typically about 300 kg. cm2 for most racquets.

5. Measuring Racquet Power

Racquet power can be measured in terms of quantity called the ACOR which stands for Apparent Coefficient of Restitution. The ACOR itself is defined as the ratio of outgoing ball speed to incident ball speed for a ball incident on a racquet initially at rest. A simple measurement method is shown in the ACOR movie where I have filmed a ball incident at low speed on a racquet hanging freely in a metal frame. There are 7 hooks at the top of the ball pendulum, spaced 20 mm apart horizontally, so the ball can impact different points on the strings from the centre to the edge of the racquet. The hooks are mounted on a horizontal metal bar that can be moved to 16 different positions spaced 20 mm apart in the vertical direction. That way, it is possible to map out the variation of ACOR over the whole string plane or to map out vibrations in the handle (with a piezo disk attached to the handle).

The film was taken from an angle to see the setup more clearly. To measure the incident and rebound ball speed, the camera would be located so it views at right angles to the path of the ball. The ACOR does NOT depend on whether the top end of the handle is supported by a length of string (as in this movie) or whether it is hinged or whether it is clamped. The ball rebounds before the reflected wave off the handle gets back to the impact point, so the ball has no way of knowing how the handle was supported.

Click photo or here to load ACOR movie. Photo shows hooks to support the top end of the ball pendulum. The “3-2-1-Go” countdown helps me to find and download the portion of the film that I want to analyse.

Results of some of these measurements.

6. Measuring Court Speed

Court speed is measured officially by the ITF using a Sestee machine. For research or club purposes, the speed of a court or any other playing surface can be measured by filming the bounce of a ball with a video camera. Some tips and various calculations required for such a measurement are outlined in a pdf file that you can download. See also the attached 0.5Mb movie or avi file showing a bounce on Rebound Ace, the hardcourt surface that used to be used at the Australian Open. The top surface is green acrylic paint mixed with fine sand to give a slightly rough surface. Underneath is a layer of rubber to soften the impact under foot.

The movie is split into a top and a bottom half. Each half is recorded at 25 frames/sec (40ms between each image) but the top half is recorded 10 ms before the bottom half, and each half is further split in half to give 50 fields/sec which makes it easier to determine when the ball bounced and to measure the spin of the ball. In other words there are 100 images/sec. The ball was incident at about 10 m/s with a small amount of backspin. It is not necessary to measure ball spin to measure court speed and bounce, but it helps to determine whether the ball was sliding throughout the bounce (as required) or whether it gripped the court at some stage during the bounce.

7. The Tennis Racquet Theorem

Hold a tennis racquet horizontally, toss it in the air so it rotates once around a horizontal axis, and catch it by the handle. The racquet will land upside-down, as shown in the movie. In the movie I taped a small red ping-pong ball on one side of the racquet so it would be easier to see what was going on and I filmed the event on the University grass courts with the physics building in the background. The flipping-over effect is described by the “Tennis Racquet Theorem”. A slow motion movie version can be viewed here.

Any object has three perpendicular axes in the x, y and z directions. In general, the moment of inertia (I) about each axis is different, with Ix > Iy > Iz. If an object starts rotating about the x or z axis it will continue to rotate about that axis without flipping over. But if it starts to rotate about the y axis then it will end up spinning about both the x and z axes as well.

It is the same with a book or a packet of cornflakes. Only one of the three axes causes flipping, and it is the one not with the smallest or the largest swing weight but the one with the medium size swing weight. Flipping over can be good or bad depending on circumstances. If the racquet happens to be a cat falling several floors out a window, this flipping over effect is good for the cat. If the racquet happens to be a fresh piece of buttered toast with jam on top, then the flipping over effect is bad, especially for the carpet. If you are an Olympic diver or gymnast, flipping over several times while doing a few somersaults can earn you extra points, perhaps leading to fame and fortune. If you are a tennis player, then watch out for the racquet twisting in your hand as you swing it. That could be good or bad depending on whether you want it to twist or not.

8. Double Pendulum Experiments

A double pendulum consists of one pendulum mounted below another so that both can swing together but the motion of one influences the other. If left to swing for a while the motion becomes chaotic, but the first half cycle is quite reproducible. Common everyday examples include the upper and lower arms and the upper and lower legs of a person, or a bat or racquet or club connected to a human arm. The swing of a bat or club or racquet can therefore be modelled by considering the motion of a double pendulum. The three movies below show how this can be done experimentally for cases where

(a) a torque is applied to help swing the upper pendulum (to simulate the effect of a muscle torque) See Movie (0.4Mb)

(b) the motion of each pendulum is slowed to have a much longer period of oscillation by mounting each arm near its centre of mass. See Movie (0.5Mb). This makes it easier to analyse film taken at only 25 or 30 frames/sec.

(c) a near perfect bat (or club or racquet) can be designed so that nearly all of the energy in the arm and the bat ends up in the ball, without any follow-through of the bat or the arm. See Movie (0.5Mb). This requires a change to the rules of the game, unfortunately, to allow for heavier balls (around 1 lb).

9. Weird tennis action

Why do players stretch the fingers of their free hand? Don’t ask me, I don’t know. Is it to increase the moment of inertia of the hand or the arm to keep it steady? My guess is that when players tense their muscles to hit a shot, they tense all their muscles, not just the ones they need to tense (about 95% of them).